boiling heat transfer - annurev.fluid.30.1

P1: ARS/ary P2: ARK/plb QC: MBL/bsa T1: MBL

November 13, 1997 13:18 Annual Reviews AR049-13

Annu. Rev. Fluid Mech. 1998. 30:365–401Copyright c© 1998 by Annual Reviews Inc. All rights reserved

BOILING HEAT TRANSFER

V. K. DhirMechanical and Aerospace Engineering Department, University of California,Los Angeles, California 90095-1597; e-mail: [email protected]

KEY WORDS: nucleate boiling, maximum heat flux, transition boiling, film boiling, minimumheat flux, pool boiling, flow boiling

ABSTRACT

This review examines recent advances made in predicting boiling heat fluxes,including some key results from the past. The topics covered are nucleate boiling,maximum heat flux, transition boiling, and film boiling. The review focuses onpool boiling of pure liquids, but flow boiling is also discussed briefly.

INTRODUCTION

Boiling is a phase change process in which vapor bubbles are formed either on aheated surface or in a superheated liquid layer adjacent to the heated surface. Itdiffers from evaporation at predetermined vapor/gas-liquid interfaces because italso involves creation of these interfaces at discrete sites on the heated surface.Nucleate boiling is a very efficient mode of heat transfer. It is used in variousenergy conversion and heat exchange systems and in cooling of high-energy–density electronic components. Pool boiling refers to boiling under naturalconvection conditions, whereas in forced flow boiling, liquid flow over theheater surface is imposed by external means. Forced flow boiling includesexternal and internal flow boiling. In external boiling, liquid flow occurs overunconfined heated surfaces, whereas internal flow boiling refers to flow insidetubes.

This review is a follow-up to Rohsenow’s (1971) similar review in this series.Reviews covering different aspects of boiling have appeared elsewhere (e.g.Kenning 1977, Dhir 1991, Fujita 1992). This review focuses on pool boilingof pure liquids. Flow boiling is described only briefly.

Boiling is a complex and elusive process. As such, we often rely on dimen-sionless groups and empirical constants when correlating data. Concurrent withthe development of correlations useful for engineering applications, progress

3650066-4189/98/0115-0365$08.00

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366 DHIR

continues to be made in understanding the physics of the boiling process. Be-cause the process is so complex and because so many heater and fluid variablesinteract, completely theoretical models have not been developed to predict theboiling heat fluxes as a function of heater surface superheat. In many cases, aconsensus is lacking in the technical community with respect to the dominantmechanisms of heat transfer (in nucleate and transition boiling) and the degreeto which the contribution of various mechanisms to total heat flux changes withwall superheat and heater geometry.

Figure 1 shows, qualitatively, the boiling curve (i.e. dependence of the wallheat flux,q, on the wall superheat on a surface submerged in a pool of saturatedliquid). The wall superheat,1T, is defined as the difference between the walltemperature and the saturation temperature of the liquid at the system pressure.The plotted curve is for a flat plate or a horizontal wire to which the heatinput rate is controlled. As the rate of heat input to the surface is increased,natural convection is the first mode of heat transfer to appear in a gravitationalfield.

At a certain value of the wall superheat (Point A), vapor bubbles appear onthe heater surface. This is the onset of nucleate boiling. The bubbles form oncavities or scratches on the surface that contain preexisting gas/vapor nuclei.In liquids that wet the surface well, the onset of nucleation may be delayed.For these liquids, a sudden inception of a large number of cavities at a certainwall superheat causes a reduction in the surface temperature, while the heatflux remains constant. This behavior is not observed when the boiling curveis obtained by reducing the heat flux, and hysteresis results. After inception,a dramatic increase in the slope of the boiling curve is observed. In partialnucleate boiling, corresponding to region II (curve AB) in Figure 1, discretebubbles are released from randomly located active sites on the heater surface.The density of active sites and the frequency of bubble release increases withwall superheat.

The transition from isolated bubbles to fully developed nucleate boiling (re-gion III) occurs when bubbles at a given site begin to merge in the verticaldirection. Vapor appears to leave the heater in the form of jets. The conditionof jet formation also approximately coincides with the merger of vapor bubblesat the neighboring sites. After lateral merger, vapor structures appear that looklike mushrooms with several stems (Gaertner 1965). Figure 2 shows a photo-graph of a large vapor structure supported by several smaller bubbles (stems).A small change in the slope of the boiling curve can occur upon transition frompartial to fully developed nucleate boiling. The heat flux on polished surfacesvaries with wall superheat roughly as

q ∼ 1Tm, (1)

wherem has a value between 3 and 4.

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BOILING HEAT TRANSFER 367

Figure 1 Typical boiling curve, showing qualitatively the dependence of the wall heat flux,q,on the wall superheat,1T, defined as the difference between the wall temperature,Tw, and thesaturation temperature,Tsat, of the liquid. Schematic drawings show the boiling process in regionsI–V. These regions and the transition points A–E are discussed in the text.

The maximum or critical heat flux,qmax, sets the upper limit of fully developednucleate boiling for safe operation of equipment. After maximum heat flux isreached, most of the surface is rapidly covered with vapor. The surface is nearlyinsulated, and the surface temperature rises very rapidly. When the rate of heatinput is controlled, the heater surface passes quickly through regions IV andV (see Figure 1) and stabilizes at point E. If the temperature at E exceeds themelting temperature of the heater material, the heater will fail (burn out). Thecurve ED (region V) represents stable film boiling, and the system can be madeto follow this curve by reducing the heat flux.

In stable film boiling, the surface is covered with vapor film, and liquiddoes not contact the solid. On a horizontal surface the vapor release pattern is

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368 DHIR

Figure 2 Photographic observation of vapor phase structure.

governed by Taylor instability of the vapor-liquid interface. With reduction ofheat flux in film boiling, a condition is reached when a stable vapor film on theheater can no longer be sustained. Heat flux and wall superheat correspondingto the condition at which vapor film collapse occurs are referred to as theminimum heat fluxqmin, and the minimum wall superheat1Tmin, respectively.

Upon collapse of the vapor film, the surface goes through regions IV, III,and II very rapidly and settles in nucleate boiling. Region IV, falling betweennucleate and film boiling, is called transition boiling, which is a mixed mode ofboiling that has features of both nucleate and film boiling. Transition boilingis very unstable, since it is accompanied by a reduction in the heat flux withan increase in the wall superheat. As a result, it is difficult to obtain steady-state data in transition boiling, except when the heater surface temperature iscontrolled. Transient transition boiling data can be obtained either by quenchingor by accessing from the nucleate boiling side when heat input to the heater iscontrolled.

NUCLEATE BOILING

Preexisting NucleiVapor/gas trapped in imperfections such as cavities and scratches on the heatedsurface serve as nuclei for bubbles. Bankoff (1958) was the first to provide

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a criterion for entrapment of gas in a wedge by an advancing liquid front.According to this criterion, a wedge-shaped imperfection on a surface will trapvapor/gas as long as the contact angle1 is greater than the wedge angle.

Wang & Dhir (1993a) developed a vapor/gas entrapment criterion by mini-mizing the Helmholtz free energy of a system involving a liquid-gas interfacein a cavity. According to this criterion, a cavity will trap vapor/gas if

φ > ψmin, (2)

whereψmin is the minimum cavity-side angle of a spherical, conical, or sinu-soidal cavity. For the spherical and conical cavities,ψmin occurs at the mouthof the cavity and is equal to the cavity-mouth angle,ψm, as measured from theheater surface. Ward & Forest (1976), while analyzing the relation betweenplatelet adhesion and roughness of a synthetic material, obtained the same cri-terion for stability of a vapor/gas nucleus in a long narrow fissure. AlthoughBankoff’s criterion provides a necessary condition for vapor/gas entrapment ina wedge, Equation 2 provides a sufficient condition.

InceptionSeveral approaches have been proposed for determining the incipient wall su-perheat for boiling from preexisting nuclei. This review discusses two of themost commonly used approaches. In the first approach, as originally proposedby Hsu (1962), an embryo will become a bubble if the temperature of theliquid at the tip of the embryo (the farthest point from the heated wall) is atleast equal to the saturation temperature corresponding to vapor pressure in thebubble. Thus, Hsu’s criterion requires that the embryo should be surroundedeverywhere by superheated liquid.

In the second approach, boiling incipience is proposed to correspond toa critical point of instability of the vapor-liquid interface. The interface isconsidered to be stable or quasi-stable if the curvature of the interface increaseswith an increase in vapor volume (see e.g. Mizukami 1977, Forest 1982, Nishio1985). Wang & Dhir (1993a) studied the instability of the vapor-liquid interfacein a spherical cavity and showed that nucleation occurs when nondimensionalcurvature of the interface attains a maximum value. They obtained the followingrelation between wall superheat and diameter,Dc, of a nucleating cavity:

1T = 4σTsat

ρvh f gDcKmax, (3)

1A distinction must be made between an advancing and a receding contact angle. Generally, theadvancing contact angle is greater than the receding contact angle. Because of the large uncertaintiesassociated with determination of advancing and receding contact angles, a static contact angle,φ,was used in this work.

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370 DHIR

where

Kmax= 1 for φ ≤ 90◦

= sinφ for φ > 90◦

In Equation 3,σ is the interfacial tension,Tsat is the saturation temperature,ρv is the density of vapor,hfg is the latent heat of vaporization, andDc is thecavity-mouth diameter. The implicit assumption made in arriving at Equation 3is that the interface temperature is the same as the wall temperature. Throughcarefully conducted experiments, Wang (1992) validated Equation 3.

According to Equation 3, few preexisting vapor/gas nuclei are found for wellwetting liquids such as R-113 and FC-72. For these liquids, the expected wallsuperheat at nucleation should approach the homogeneous nucleation temper-ature ('90% of critical temperature). The observed inception superheats forthese liquids, although much higher than those observed for partially wettingliquids, are much smaller than those corresponding to homogeneous nucleationtemperature (see Barthau 1992). Gases dissolved in these liquids may initi-ate the nucleation, and as a result, the observed superheat is smaller than thatcorresponding to homogeneous nucleation. In many instances gas is added byexternal means to the wetting liquids to reduce the inception temperature andto minimize the hysteresis.

Nucleation Site DensityThe number density of sites that become active increases as wall heat flux orsuperheat increases. Because addition of new nucleation sites influences therate of heat transfer from the surface, a knowledge of nucleation site density asa function of wall superheat is needed in order to develop a credible model forprediction of nucleate-boiling heat flux. Several other parameters also affectthe site density, including the procedure used in preparing the heater surface,surface finish, surface wettability, heater material thermophysical properties,and heater thickness. Until recently, little attention had been given to the effectof these parameters on the density of active sites. Kocamustafaogullari & Ishii(1983) correlated the cumulative nucleation site density reported by variousinvestigators for water boiling on a variety of surfaces at pressures of 1–198atm. In developing the correlation, heater surface characteristics were notconsidered. In general, active site density is correlated as

Na ∼ 1Tm1, (4)

wherem1 varies between 4 and 6. Cornwell & Brown (1978) found that theproportionality constant in Equation 4 increased with surface roughness, butthe exponent was independent of surface roughness. Bier et al (1978) obtained

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distinctly different values ofm1 during boiling on an etched copper surface andon a turned surface. This discrepancy resulted because in neither case were theobservations tied to the shape and size distribution of cavities.

Wang & Dhir (1993a,b) provided a mechanistic approach for relating thecavities present on the surface to those that actually nucleate. Their approachalso includes the effect of surface wettability. They first determined the size,shape, and mouth angle of activities present on a polished copper surface andthen used Equation 2 to determine the fraction of those cavities that will trapvapor/gas. Most of the cavities that could trap vapor/gas were of reservoir type.The data, consistent with their model, showed a 20-fold reduction in numberdensity of active sites as the contact angle was decreased from 90◦ to 18◦.Although Wang & Dhir showed how the developed criterion could be used todetermine theoretically the number density of active sites, the procedure usedin determining the size, shape, and mouth angle of cavities is tedious and timeconsuming and cannot be used readily in a practical application.

Wang & Dhir did not consider the thermal interference between sites orthe seeding and deactivation of sites in the neighborhood of an active cavity.Kenning (1989) noted that thermal and flow conditions in the vicinity of aheated surface can lead to activation of inactive sites and deactivation of activesites. Sultan & Judd (1983) studied the bubble growth pattern at neighboringsites during nucleate pool boiling of water on a copper surface. They found thatelapsed time between the start of bubble growth at two neighboring active sitesincreased as the distance separating the two sites increased. They proposedthat thermal diffusion in the substrate in the immediate vicinity of the boilingsurface may be responsible for this behavior. Their work suggests that somerelation may exist between distribution of active nucleation sites and bubblenucleation phenomenon.

Judd & Chopra (1993) reported results of interactions between neighboringsites that lead to activation of inactive sites and deactivation of active sites. Theynoted that for separation distances between nucleation sites less than one bubblediameter at departure, the formation of a bubble at the initiating site promotesthe formation of bubbles at the adjacent sites (site seeding). For separationdistances between one and three bubble diameters at departure, formation ofa bubble at the initiating site inhibits the formation of bubbles at the adjacentsite (deactivation of sites). For distances greater than three bubble diameters atdeparture, nucleation at one location is not influenced by activation at anothersite. Kenning’s and Judd’s studies indicate that thermal interference and siteseeding or deactivation can alter the local active-site density and distributionat low heat fluxes or in partial nucleate boiling. However, the significance ofthese processes with respect to heat transfer during well-established nucleateboiling on thick heaters is expected to be small.

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372 DHIR

Bubble DynamicsAfter inception, a bubble continues to grow (in a saturated liquid) until forcescausing it to detach from the surface exceed those pushing the bubble againstthe wall. After departure, cooler liquid from the bulk fills the space vacatedby the bubble, and the thermal layer at and around the nucleation site reforms(transient conduction). When the required superheat is attained at the tip ofthe vapor bubble embryo or the interface instability criterion is met, a newbubble starts to form at the same nucleation site, and the bubble growth processrepeats. Wall heat transfer in nucleate boiling results from natural convection onthe heater surface areas not occupied by bubbles and from transient conductionand evaporation at and around nucleation sites. Bubble dynamics include theprocesses of bubble growth, bubble departure, and bubble release frequency,which includes time for reformation of the thermal layer (waiting period). Thefollowing sections describe each of these processes.

BUBBLE GROWTH The literature highlights two points of view with respect tobubble growth on a heated surface. One group of investigators has proposedthat the growth of a vapor bubble occurs as a result of evaporation all around thebubble interface. The energy for evaporation is supplied from the superheatedliquid layer that surrounds the bubble after its inception. Some bubble growthmodels are similar to that proposed for growth of a vapor bubble in a sea ofsuperheated liquid (see Plesset & Zwick 1954). The bubble growth process ona heater surface, however, is more complex because the bubble shape changescontinuously during the growth process, and superheated liquid is confined to athin region around the bubble. Mikic et al (1970) obtained an analytical solutionfor the bubble growth rate by using a geometric factor to relate the shape of abubble growing on the heater surface to a perfect sphere and accounting for thethermal energy stored in the superheated liquid layer prior to bubble inception.Since the initial energy content of the superheated liquid layer surrounding thebubble depends on the waiting time, the model shows the dependence of bubblegrowth rate on waiting time.

The second point of view is that most of the evaporation occurs at the baseof the bubble in that the microlayer between the vapor-liquid interface andthe heater surface plays an important role. Snyder & Edwards (1956) werethe first to propose this mechanism for evaporation. Moore & Mesler (1961)deduced the existence of a microlayer under the bubble from the oscillations inthe temperature measured at the bubble release site. Cooper & Lloyd (1969)not only confirmed the existence of a microlayer underneath isolated bubblesformed on glass or ceramic surfaces but also deduced the thickness of themicrolayer from the observed response of the heater surface thermocouple.They noted that an expression for local thickness,δ, of the microlayer could be

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written as

δ ∼√νl tg, (5)

whereν l is the kinematic viscosity of the liquid, andtg is the bubble growth time.It was further demonstrated that bubble growth was mostly due to evaporationfrom the microlayer. Although Cooper & Lloyd’s work proved the importanceof microlayer evaporation at low pressures, their work was limited in scope. Lee& Nydahl (1989) calculated the growth of spherical bubbles with a microlayer.For microlayer thickness, they used Cooper & Lloyd’s formulation. They cameto the same conclusion as Cooper & Lloyd, that microlayer evaporation is asignificant contibutor to the heat transfer during bubble growth. However,Plesset & Prosperetti (1977) concluded that in subcooled boiling, evaporationat the microlayer accounts for only 20% of the total heat flux. After morethan three decades of research, we still do not have an effective, consistentmodel for bubble growth on a heated surface that appropriately includes themicrolayer contribution and time-varying temperature and flow field around thebubble.

BUBBLE DEPARTURE The diameter to which a bubble grows before departingis dictated by the balance of forces that act on the bubble. These forces areassociated with the inertia of the liquid and vapor, the liquid drag on the bubble,buoyancy, and the surface tension. Fritz (1935) correlated the bubble departurediameter by balancing, on a static bubble, the buoyancy with surface tensionforce. Although significant deviations of the bubble diameter at departure withrespect to Fritz’s expression have been reported in the literature, especially athigh system pressures, his correlation did provide a correct length scale for theboiling process.

Several other expressions have been reported for bubble diameter at depar-ture, obtained either empirically or analytically by involving various forcesacting on a bubble. These expressions (see e.g. Hsu & Graham 1976), how-ever, are not always consistent with each other. Cole & Rohsenow (1969)correlated bubble diameter at departure with fluid properties but found it to beindependent of wall superheat. Gorenflow et al (1986) proposed an expressionfor bubble diameter at departure that indicates the bubble diameter increasesweakly with wall superheat.

Some investigators disagree about the role of surface tension. Generally,surface tension tends to push the bubble against the wall and thus inhibits bubbledeparture. However, Cooper et al (1978) found that in some cases surfacetension assisted bubble departure by making the bubble spherical. Buyevich &Webber (1996) also made the same argument. Issues regarding the forces thatact on a growing bubble can be put to rest only through complete numerical

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374 DHIR

simulation of both bubble growth and departure while properly accounting forthe adhesion forces and interfacial tension.

BUBBLE RELEASE FREQUENCY A theoretical evaluation of the bubble releasefrequencyf can be made from the expressions for the waiting timetw andthe growth timetg. The waiting time corresponds to the time it takes for thethermal layer to redevelop to allow nucleation of a bubble. Predictions of bubblerelease frequency based on waiting and growth times, however, do not matchwell with the data because many simplifications are made in obtainingtw andtg. Thus, correlations have been reported in the literature that include both thebubble diameter at departure and bubble release frequency. One of the mostcomprehensive correlations of this type is given by Malenkov (1971).

Heat Transfer MechanismsIn partial nucleate boiling, or in the isolated bubble regime, transient conductioninto liquid adjacent to the wall is an important mechanism for heat transferfrom an upward-facing horizontal surface (Forster & Greif 1959). After bubbleinception, the superheated liquid layer is pushed outward and mixes with thebulk liquid. The bubble acts like a pump in removing hot liquid from the surfaceand replacing it with cold liquid. Combining the contribution of transientconduction on and around nucleation sites, microlayer evaporation underneaththe bubbles, and natural convection on inactive areas of the heater, one canwrite an expression for partial nucleate-boiling heat flux as

q = K 2

2

√π(kρcp)l f D2

d Na1T +(

1− K 2

2NaπD2

d

)hnc1T

+ hev1T Naπ

4D2

d. (6)

In the above equationkl is the thermal conductivity of liquid,ρ l is the densityof liquid, andcρl is the specific heat of liquid.

Only the first two terms in Equation 6 were included in Mikic & Rohsenow’s(1969) original model. The evaporation at the bubble boundary is includedin the first term that represents the transient conduction in the liquid. Judd &Hwang (1976) suggested the addition of the last term on the right-hand side ofEquation 6. This term accounts for the microlayer evaporation at the base ofbubbles. For Equation 6 to serve as a predictive tool, several variables mustbe known: the bubble diameter at departure,Dd; bubble release frequency,f,the proportionality constant,K, for the bubble diameter of influence; numberdensity,Na, of active sites; and average heat transfer coefficients,hnc andhev, fornatural convection and microlayer evaporation, respectively. Using empiricalcorrelations for several of these parameters, Mikic & Rohsenow (1969) justified

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the validity of Equation 6 when the third term on the right-hand side of theequation was not included.

Judd & Hwang (1976) matched the heat fluxes predicted from Equation 6with those observed in the experiments in which dichloromethane was boiledon a glass surface. In doing so, they relied on the measured values of microlayerthickness to evaluatehev and on the assumption thatK2 was 1.8. Experimentallymeasured values of active nucleation site density and bubble release frequencywere used in the model. Figure 3 shows Judd & Hwang’s data and predictions.At the total measured heat flux of 6 w/cm2, about one third of the energy isdissipated through evaporation at the bubble base. The data plotted in Figure 3

Figure 3 Relative contribution of various mechanisms to nucleate-boiling heat flux (Judd &Hwang 1976).qM, measured heat flux;qP, predicted heat flux;qME, microlayer evaporation heatflux; qNC, natural convection heat flux.

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376 DHIR

show that at high heat fluxes or in fully developed nucleate boiling, most ofthe energy from the heater is removed by evaporation. This observation is ingeneral agreement with Gaertner’s (1965) finding that after the first transition(partial to fully developed nucleate boiling), evaporation is the dominant modeof heat transfer.

At low heat fluxes, or in partial nucleate boiling, the relative contributionof various mechanisms depends on the geometry of the heater. In fact, thedetails of the heat transfer mechanisms may be altered as heater geometry orthe angular position of the surface with respect to the direction of gravitationalacceleration is varied. For example, on a downward-facing surface, the bubblesmay slide along the heater surface for some distance after leaving the nucleationsite but before moving away from the heater surface. During the movementof the bubbles along the heater surface, cyclic disruption and reformation ofthe thermal layer will occur and, in turn, will result in a higher heat transferrate. Figure 4 shows the nucleate boiling data obtained by Nishikawa et al(1974) for water on flat plates inclined at different angles with the horizontal. Inpartial nucleate boiling, the downward-facing surfaces accommodate heat fluxesthat are higher than those on an upward-facing horizontal surface or a verticalsurface. However, at high heat fluxes or in fully developed nucleate boiling,the data for all of the surfaces fall on a single line. This observation indicatesthat when evaporation is the dominant mode of heat transfer, the orientation ofthe plate has little effect on dependence of heat flux on wall superheat.

In fully developed nucleate boiling, mushroom-type bubbles supported byseveral vapor stems attached to the heater may be observed (Gaertner 1965;see also Figure 2). Most evaporation occurs at the periphery of these stems(smaller bubbles supporting large vapor masses). Energy for the phase changeis supplied by the superheated liquid layer in which the stems are implanted.Thus, the boiling heat flux can be calculated if the fractional area occupied bythe vapor stems and the thickness of the thermal layer are known. The heaterarea fraction occupied by the vapor stems is equal to the product of the numberdensity of stems and the wall area occupied by one stem.

Alternatively, the heat flux can be calculated if the vaporization rate per stemand number density of active sites are known. Lay & Dhir (1994) used the latterapproach to predict fully developed nucleate-boiling heat flux. By assumingthat the duration for which vapor stems exist on the heater is much larger thanthe time needed to form the stems, Lay & Dhir (1995a) carried out a quasi-staticanalysis to determine the maximum diameter of vapor stems as a function ofwall superheat. The shape of the vapor stem depends on the value chosen forthe Hamaker constant.

Lay & Dhir’s analysis also showed that locally, in the ultra-thin film, heatfluxes as high as 1.54× 108 W/m2 could exist at a wall superheat of 20 degreesK for water at 1 atmospheric pressure. This observation is worthy of further

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Figure 4 Nucleate boiling data of Nishikawa et al (1974) on plates oriented at different angles tothe horizontal.

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378 DHIR

exploration in our pursuit to accommodate very high heat fluxes at relativelylow wall superheats. From the analysis, the vaporization rate,ms, per stem canbe calculated as a function of wall superheat. Using Wang & Dhir’s (1993a,b)model for density of active sites, the heat flux in fully developed nucleate boilingwas calculated from

q = Namsh f g. (7)

Heat fluxes predicted from Equation 7 agreed well with the data. In fact,good agreement with Gaertner & Westwater’s (1960) data was also seen whennumber density of active sites reported by Gaertner & Westwater was used.This approach needs to be verified further with data from other sources.

Heat Transfer CorrelationsBecause mechanistic models are lacking for several parameters (e.g.Na, Dd,f, hev), prediction of heat flux from Equation 6 requires adjustment of severalempirical constants embedded in these parameters. As a result, Equation 6, pre-sumably obtained on mechanistic arguments, cannot be readily used to predictthe dependence of nucleate-boiling heat flux on wall superheat. Most often,correlations reported in the literature have been used for this purpose. Thesecorrelations generally are valid for both partial and fully developed nucleateboiling. For example, Rohsenow’s (1952) correlation has been used widely,even though it is not based on correct physics. Stephan & Abdelsalam (1980)developed a comprehensive correlation for saturated nucleate pool boiling ofdifferent liquids. Their correlation is based on both fluid and solid properties,but no consideration is given to heater geometry.

Cooper (1984a,b) proposed a simple correlation for saturated nucleate poolboiling. His correlation uses reduced pressure, molecular weight, and surfaceroughness as the correlating parameters. His correlation for a flat plate can bewritten as

(q)1/3

1T= 55.0

(p

pc

)0.12−0.21 log10 Rp

·(− log10

p

pc

)−0.55

· M−0.50. (8)

In Equation 8, the roughness,Rp, is measured in microns,M is the molecularweight, p is the system pressure,pc is the critical pressure,1T is measuredin degrees K, andq is given in W/m2. Cooper suggested that for applicationof the correlation to horizontal cylinders, the lead constant on the right-handside should be increased to 95. Equation 8 accounts for roughness but does notaccount for variations in degree of surface wettability. These correlations shouldbe used with caution, as large deviations between actual data and predictionscan occur when the conditions under which the correlation was developed arenot duplicated.

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Effect of System VariablesSeveral system variables such as surface finish, surface wettability, surface con-tamination, heater geometry, liquid subcooling, flow velocity, gravity, systempressure, thermal properties of the solid, and the mode in which the tests areperformed influence the dependence of nucleate-boiling heat flux on wall super-heat. For example, surface roughness pushes the boiling curve to the left. Im-proved wettability suppresses nucleation and, as a result, shifts the boiling curveto the right. Physicochemical changes on the surface can take place because ofdeposition of inert matter contained in the host liquid, slow chemical reactionof the surface with the gases dissolved in the liquid or with the vapor, and strongchemical reaction of the metal with the concentrated solutions of electrolytes.Generally, the effect of surface contamination is to enhance the wettability andthereby reduce the nucleate-boiling heat flux for a given wall superheat.

As noted from Figure 4, partial nucleate-boiling heat fluxes generally arehigher on a downward-facing surface, but in fully developed nucleate boilingthe surface orientation has little effect. Thus, the geometry of the surface canhave an effect on partial nucleate-boiling heat fluxes. The rate of convectiveheat transfer increases with liquid subcooling. As a result, liquid subcool-ing influences the inception and partial nucleate boiling regions of the boilingcurve. On the wall heat flux vs wall superheat plots, convective and partialnucleate-boiling heat fluxes for subcooled liquids lie higher than those for sat-urated boiling. However, at high nucleate-boiling heat fluxes, the subcooledand saturated boiling curves almost overlap. Similarly, flow velocity enhancesconvective and partial nucleate-boiling heat fluxes but has little effect on fullydeveloped nucleate boiling.

The magnitude and direction of gravitational acceleration with respect tothe heater surface influences the hydrodynamic and thermal boundary layersand bubble trajectory. In partial nucleate boiling, heat transfer by convectionrepresents a major fraction of the total heat transfer rate. Thus, gravity playsan important role in this mode of boiling. However, Merte’s (1988) centrifugedata and Zell et al’s (1989) low-gravity data showed that the magnitude ofgravity has little effect on fully developed nucleate boiling. With an increasein system pressure, the incipience superheat decreases and the nucleate boilingcurve shifts to the left.

The nucleate-boiling heat transfer data collected by Stephan & Abdelsalam(1980) suggested that thermophysical properties of the solid can have a weakeffect on nucleate-boiling heat fluxes. The boiling curve can be affected bythe manner in which the heat flux is imposed on the surface—steady state ortransient. Sakurai & Shiotsu’s (1977a,b) experiments on platinum wires sub-merged in a pool of saturated water showed that for exponential heating periods

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380 DHIR

varying from 5 ms to 1 s, the incipient heat flux increases as the exponentialtime decreases. In nucleate boiling, the transient heat transfer coefficients gen-erally are lower than those obtained under steady-state conditions. The ratioof transient and steady-state heat fluxes depends on the magnitude of the heatflux, but this ratio can be as low as 0.5.

Heat Transfer EnhancementThe size of equipment needed for a given heat load can be reduced if alreadyhigh nucleate-boiling heat fluxes can be further enhanced. Many research ef-forts have been made in this direction. Recent advances in enhancing nucleate-boiling heat flux have included the development of heater surfaces with high-density interconnected artificial cavities of the re-entrant type. Figure 5 showsa reentrant-type cavity and the structures of two of the commercially availablesurfaces. The cavities on these surfaces nucleate at very low superheats. Thenucleate-boiling heat fluxes are enhanced not only by the high density of activenucleation sites at low superheats but also by the evaporation of a thin liquidfilm formed on the cavity walls. The film results from the liquid that is pushedinto the cavity after a bubble leaves. The enhanced surfaces have led to anorder of magnitude increase in already high nucleate-boiling heat transfer co-efficients. However, enhancement is much less at high wall superheats or nearthe maximum heat flux condition.

MAXIMUM HEAT FLUX

The maximum or critical heat flux represents the upper limit of nucleate-boilingheat flux and marks the termination of efficient cooling conditions on the sur-face. Several experimental and theoretical studies have been reported thatdelineate the physics of onset of the critical heat flux condition in pool boiling.However, no clear consensus exists in the technical community as to the actualmechanism of critical heat flux.

Figure 5 Reentrant cavity and commercially available enhanced surfaces. (a) Cross-section ofreentrant-type cavity. (b) Thermal excel-E surface. (c) Gewa-T surface.

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MechanismsKutateladze (1948) and Zuber (1959) proposed two early models for predictionof maximum heat flux on large horizontal surfaces. Both models are basedon the hydrodynamics of vapor outflow. Kutateladze developed dimensionlessgroups from the equations governing the flow of vapor and liquid. Zuber, incontrast, proposed that the maximum heat flux occurs when velocity in the vaporjets issuing from the surface reaches a critical velocity. The critical velocity isthe velocity at which vapor jets become Kelvin-Helmholtz unstable. Zuber alsoassumed that the jet diameter was half of the jet spacing that was bounded bythe “critical” and the “most dangerous” two-dimensional Taylor wavelengths.For inviscid liquids, Zuber obtained an expression for the maximum heat fluxon infinite flat plates as

qmaxF = Cρvh f g4

√σg(ρl − ρv)

ρ2v

(ρl + ρvρl

)1/2 [ρl (16− π)

ρl (16− π)+ ρvπ]. (9)

At low system pressures, Equation 9 and the expression obtained by Kutate-ladze are nearly identical. The value obtained by Zuber for constantC wasπ /24, whereas Kutateladze correlated the data available at that time and foundthe constantC to have a value of 0.168. Subsequently, Lienhard & Dhir (1973)obtained data with a variety of fluids at different accelerations normal to theheaters and concluded that for large horizontal plates constant,C should havea value of 0.15. From the data they also deduced that for a plate to be called alarge plate it should at least accommodate three Taylor wavelengths. Neithermodel accounted for the surface wettability, and presumably the underlyingassumption in these models was that liquids wetted the heater surface well.

Equation 9 has also been extended to predict maximum heat flux on heatersof different geometry, size, and orientation (Lienhard & Dhir 1973). For heatersof other geometries, the maximum heat flux is written as

qmax= f (l ′)qmaxF , (10)

where f (l ′) is a function of dimensionless characteristics width,l ′, of the heater,which is defined as

l ′ = l√σ

g(ρl−ρv). (11)

For small heaters,l ′ < 1, the function f (l ′) increases asl ′ decreases. Forlarge heaters, the functionf (l ′) becomes independent ofl ′ and attains a valueslightly less than unity. However, the exact value off (l ′) depends on theheater geometry. The prediction of maximum heat flux on heaters of differentgeometries requires a knowledge of the ratio of vapor jet to heater area and of

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382 DHIR

critical velocity of vapor in the jets. Lienhard & Dhir (1973) summarized themethodology for evaluatingf (l ′) for various heater geometries. Predictionsfrom Equation 10 agree with a large set of maximum heat flux data obtainedwith different liquids and heater geometries.

Haramura & Katto (1983) have questioned the validity of the assumption ofinstability of large vapor jets used in the hydrodynamic theory as originally pro-posed by Zuber and its subsequent augmentation by Lienhard & Dhir (1973).This questioning is based on the fact that visual observations show the presenceof large vapor mushroom-type bubbles on the heater surface rather than tallvapor jets. Haramura & Katto (1983) suggested an alternative hydrodynamicmodel for prediction of maximum heat flux under pool boiling conditions.In their model, the vapor stems supporting mushroom-type bubbles becomeHelmholtz unstable. Maximum heat flux is proposed to occur when the liquidfilm trapped between the base of the mushroom-type bubble and the wall driesout prior to departure of the bubble (hovering period). The thickness of theliquid film is assumed to be one fourth of the Helmholtz unstable wavelength.By comparing the maximum heat flux predicted from their model with the pre-diction from Equation 9 withC = π /24, Haramura & Katto found the ratioof vapor stem to heater area to be a function of vapor to liquid density ratioonly. However, the predicted magnitude of the area ratio is not borne out byexperiments, and the model is unable to describe the observed effect of surfacewettability on the maximum heat flux. Nevertheless, several investigators haveextended the Haramura & Katto model to flow boiling and to jet impingementcooling.

From the early 1960s to the late 1970s, the hydrodynamic theory was wellaccepted as a model of the maximum heat flux mechanism under pool boilingconditions. However, during that time period, questions persisted regarding thetheory’s ability to predict maximum heat fluxes on surfaces that were not wellwetted. The observed maximum heat fluxes on partially wetted surfaces arelower than those predicted by the hydrodynamic theory.

Only recently have several studies documented, unambiguously, the effectof surface wettability. Liaw & Dhir (1986) studied boiling of saturated water at1 atm on a vertical surface. A prescribed procedure was followed for oxidationof the surface, and the static contact angle was used as the measure of thedegree of wettability. Maracy & Winterton (1988) obtained similar data on ahorizontal plate, whereas Hahne & Disselhorst (1978) used horizontal cylindersof different materials. These investigators showed that the maximum heat fluxdecreases with increase in contact angle. However, in comparison to the dataof Liaw & Dhir and Maracy & Winterton, the data of Hahne & Disselhorstobtained on cylinders showed a much stronger dependence of maximum heatflux on contact angle.

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Figure 6 Dependence of peak heat flux on contact angle.

Figure 6 shows the steady-state peak heat flux data obtained by Liaw & Dhir(1986). Saturated water at 1 atm pressure was the test liquid. The data areplotted as a function of contact angle and were taken on a 6.3-cm wide and10.3-cm high copper plate. Also plotted are the data obtained with R-113,which wets the polished copper surface well. The dotted lines in Figure 6 showthe predictions obtained from Equation 9 using the value ofC suggested byZuber (for an infinite horizontal plate) and that suggested by Lienhard & Dhirfor a vertical plate. The data obtained with R-113 and with water at a contactangle of 18◦ are within a few percent of the prediction based on hydrodynamictheory. However, water data covering a range of contact angles from 27◦ to107◦ are much lower.

For a contact angle of 90◦ (polished copper, distilled water), the observedmaximum heat flux is only about 55% of that given by Lienhard & Dhir (1973).The reduction of maximum heat flux noted by Costello & Frea (1965) whendistilled water was used instead of tap water can thus be attributed to the reducedwettability. Lienhard & Dhir explained these data by considering the numberof vapor jets that the heater accommodated. Since the available buoyancy

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384 DHIR

force can sustain a vapor removal rate corresponding to the maximum heatflux for an 18◦ contact angle, the hydrodynamics of the vapor outflow cannotdetermine the maximum heat flux on partially wetted surfaces. In contrast,because the maximum heat flux data appear to be correlated with the surfacewettability (surface property), the upper limit of heat removal is likely set by thesurface.

Further evidence that hydrodynamics does not control the maximum heat fluxon partially wetted surfaces is obtained from the void fraction profiles. The voidfraction on partially wetted surfaces at maximum heat flux is less than unityeverywhere normal to the surface; thus, flow paths are available for the liquidto reach the heater surface. Dhir & Liaw (1989) explained the occurrence ofmaximum heat flux on partially wetted surfaces on the basis of evaporation areaavailable at the stem interface. For a contact angle of 90◦, the vapor stems mergeat the wall when the wall void fraction attains a value ofπ /4. For contact anglesless than 90◦, the merger occurs away from the wall. After merger, the stemcircumference in contact with liquid decreases rapidly. The higher the interfacearea available for evaporation at a given superheat, the higher the heat flux (Dhir& Liaw 1989). Thus, a maximum in the interfacial area corresponds to onsetof the maximum heat flux condition, or the maximum rate of evaporation at thestems sets the upper limit of nucleate-boiling heat flux. For partially wettedsurfaces, the merger of vapor stems signals a degradation in heat removal rateor onset of transition boiling. In some respects, this mechanism for maximumheat flux is similar to Rohsenow & Griffith’s (1956) proposal that onset ofthe maximum heat flux condition is due to packing of bubbles, which leads tocoalescence of isolated bubbles at the heater surface.

For surfaces with contact angles less than 90◦, maximum void fraction occursslightly away from the wall. At maximum heat flux on well-wetted surfaces(maximum heat fluxes approaching those predicted from the hydrodynamictheory), the void fraction away from the wall reaches a value of unity. At avoid fraction equal to unity, an obstruction to the flow of liquid to the wall candevelop. Since in steady state the vapor production rate must equal the vaporremoval rate, this condition appears to be analogous to what has been assumedin the past with respect to the hydrodynamically controlled boiling crisis inpool boiling. The hydrodynamic theory proposed by Zuber (1959) assumedthat the maximum heat flux occurs when vapor escape velocity and vapor flowarea fraction reach their critical values. A void fraction of unity slightly awayfrom the heater is an alternative form of the same criterion.

Thus, Dhir & Liaw (1989) have proposed different mechanisms for maximumheat flux on partially wetted and well-wetted surfaces. For partially wettedsurfaces the mechanism depends on the evaporation rate at the surface, whereasfor well-wetted surfaces the mechanism depends on the vapor removal rate or

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occurrence of the void fraction of unity (hydrodynamic limit) slightly awayfrom the wall.

Unified ApproachDhir & Liaw also provided a framework for a unified model for nucleate andtransition pool boiling on partially wetted surfaces. In their model the maxi-mum heat flux condition was not considered as a disjoint process but rather asa transition point in the continuously evolvingq−1T curve encompassing thethree modes of boiling (i.e. nucleate boiling, transition boiling, and film boil-ing). Since heat removal on both the wet and dry areas contributes to the overallheat transfer from the surface, an expression for the time- and area-averagedheat flux may be written as follows:

q = ql (1− αw)+ qvαw

≡ hl (1− αw)1T + hvαw1T. (12)

In Equation 12,ql andhl are the time- and area-averaged heat flux and heattransfer coefficient, respectively, over the liquid-occupied region. Similarly,qvandhv are the time- and area-averaged heat flux and heat transfer coefficient,respectively, on the dry region.

In Equation 12, the temperature over the dry and wet areas is assumed to bethe same. Such an assumption is true for thick copper plates. However, forthin heaters made out of low-conductivity materials, a significant difference intemperature between dry and wet areas may exist. In evaluating the heat fluxfrom Equation 12, Dhir & Liaw (1989) calculatedhl by knowing the energyremoval rate by evaporation at the periphery of vapor stems. The heat transfercoefficient,hv, over the dry region was obtained from Bui & Dhir’s (1985a)data for film boiling on a vertical surface, and experimentally measured valuesof wall void fraction were used. As discussed earlier, Wang & Dhir (1993a,b)and Lay & Dhir (1995a) provided a theoretical basis for prediction of numberdensity of active sites and the diameters of the vapor stems, respectively. If thenumber density and the dry area underneath a vapor stem are known, the wallvoid fraction can be determined.

Although Dhir & Liaw used the unified model to predict nucleate and transi-tion boiling on partially wetted surfaces, the approach can be applied to surfaceson which maximum heat flux is determined by the vapor removal limit (e.g.well-wetted surfaces). However, implementation of the model requires knowl-edge of the variation in wall void fraction with wall superheat after the vaporremoval limit has been reached. To have a totally predictive model for post–critical heat flux, one needs to have models for wall void fraction and the shapeof the interface near the heater surface.

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386 DHIR

Effect of System VariablesSeveral system variables affect the maximum heat flux: surface wettability;heater geometry, size, material, and thickness; liquid subcooling; flow velocity;gravity; system pressure; and the mode in which the surface is heated. The effectof wettability, geometry, and size of the heater on the maximum heat flux wasdiscussed earlier. Ample evidence supports the conclusion that for thin heatersmade of low-conductivity materials such as steel or inconel, the maximum heatflux is lower than that predicted from the hydrodynamic theory (well-wettedsurface). In the earlier studies, the maximum heat flux was correlated with theproduct of density, specific heat, and thickness of the heater material (Houchin& Lienhard 1966, Tachibana et al 1967).

Bar-Cohen & McNeil (1992), Carvalho & Bergles (1992), and Golobic &Bergles (1992) analyzed a large body of critical heat flux data on heaters ofdifferent materials and thickness. They correlated the critical heat flux withconpacitance (the product of the heater thickness and of the square root of theproduct of the thermal conductivity, specific heat, and density of the heatermaterial). For thick heaters, the critical heat flux appeared to asymptoticallyapproach the hydrodynamic limit. From such a correlation of the data, Carvalho& Bergles found the thickness of the heater material required to achieve at least90% of the asymptotic value of the critical heat flux.

The maximum heat flux increases with liquid subcooling. Zuber et al (1961)extended Equation 9 to a subcooled liquid by accounting for heat lost to theliquid in a transient manner during growth of a bubble. Elkassabgi & Lienhard(1988) investigated maximum heat fluxes during subcooled pool boiling onhorizontal cylinders. They identified three subcooling regimes. For low sub-coolings, the maximum heat flux varies linearly with subcooling in a mannersimilar to that observed by Zuber et al. At moderate subcoolings, bubbles sur-rounded the heater without detaching. For these subcoolings, the maximumheat flux varied slightly nonlinearly with liquid subcooling and was determinedby natural convection from the outer edge of the bubble boundary layer. Athigh subcoolings, the maximum heat flux was independent of liquid subcoolingand was limited by the evaporation rate at the heater surface (i.e. moleculareffusion limit) and not by the rate at which energy could be removed by naturalconvection from the outer edge of the bubble boundary layer.

The flow velocity enhances the maximum heat flux and is discussed later.According to Equation 9, the maximum heat flux should scale as4

√g. However,

for very low gravities (µg), the functional dependence of maximum heat fluxon gravity is weaker than that obtained from Zuber’s hydrodynamic analysis.The reasons for weaker dependence of maximum heat flux on gravity undermicrogravity conditions are not clearly understood. Questions also remainabout the stability of boiling. Merte (1994) reported that subcooled boiling

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during long periods of microgravity is unstable and that the surface alternatelywets and dries out prior to occurrence of critical heat flux.

The effect of system pressure on maximum heat flux is built into the hy-drodynamic model (Equation 9). With increase in system pressure, the criticalheat flux attains a maximum value near a reduced pressure of about 0.35. Themagnitude of the maximum heat flux is affected if the heat input to the heateris increased rapidly. Sakurai & Shiotsu (1977a,b) found that for exponentialheating periods less than 100 ms, the transient maximum and DNB heat fluxesincrease as the exponential time decreases. The DNB heat flux is defined asthe highest nucleate-boiling heat flux at which a linear relationship betweenln q and ln1T ceases to exist. Expressions for transient maximum heat fluxesusing a steady-state critical heat flux model as the starting point have been de-veloped (Serizawa 1983, Pasamehmetaglou et al 1987). Maximum heat fluxesobserved during quenching of solids are generally lower than their steady-statevalues. By carrying out quenching experiments on copper discs in liquid ni-trogen, Peyayopanakul & Westwater (1978) showed that transient maximumheat fluxes decrease as disc thickness decreases. However, for discs thickerthan 2.5 cm, the maximum heat flux is independent of thickness. If the time totraverse the top 10% of the boiling curve is greater than 1 s, the boiling processcan be called quasi-steady. Lin & Westwater (1982) showed that similar tosteady-state experiments, the heater thickness and thermophysical propertiesof the heater have some influence on the boiling curve, including maximumheat flux obtained during quenching.

TRANSITION BOILING

Transition boiling is characterized by a reduction in surface heat flux with anincrease in wall superheat. As a result, the process is inherently unstable. Intransition boiling, periods of liquid-solid and vapor-solid contact occur alterna-tively at a given location on the heated surface. Conditions similar to nucleateboiling and film boiling prevail during wet and dry periods respectively. Thevariation in heat flux with wall superheat is a result of change in the fractionof time each boiling mode is present on a given area. Reviews on transitionboiling have been presented (e.g. Dhir 1991, Auracher 1992), and major pointsfrom these reviews and some recent results are included here.

Since Witte & Lienhard (1982) asserted that Berenson’s (1962) transitionboiling data showed hysteresis, several studies on the issue have appeared inthe literature. Bui & Dhir (1985b) showed from their experiments on a verticalsurface that different transition boiling curves were obtained depending onthe side of the curve from which the boiling was accessed (i.e. the nucleateboiling side or the film boiling side). Bui & Dhir (1985b) and Liaw & Dhir

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(1986) showed that the magnitude of hysteresis in transition boiling curvesdepends on the static contact angle. The larger the static contact angle, the largerthe hysteresis. In fact, for R-113, which wets the surface well, no hysteresiswas observed when transition boiling was accessed either from the film boilingside or from the nucleate boiling side. However, Rajab & Winterton (1990)claim that in their experiments on a horizontal surface, hysteresis persisted evenwhen liquid wetted the surface well (i.e. nearly zero contact angle).

Ramilison & Lienhard (1987) recreated Berenson’s apparatus in which steamwas condensed on the underside of a copper disc while boiling occurred onthe top of the disc. In this experiment, although thermal resistance of thecopper heater was reduced, not all points in transition boiling were accessible.Ramilison & Lienhard conjectured that the shift from film-transition boilingto nucleate-transition boiling was a result of the change from an advancingcontact angle to a receding contact angle. Thus, they implied that hysteresiswas a result of differences in advancing and receding contact angles.

Haramura’s (1991) data, also obtained on an apparatus similar to that ofBerenson, showed that for R-113, some hysteresis existed even though the datawere obtained under steady-state conditions. Auracher (1992) developed afeedback system so that in electrically heated systems, steady-state transitionboiling data could be obtained. He found that no hysteresis existed during tran-sition boiling of R-113 when data were obtained under steady-state conditionsunder either increasing or decreasing temperature conditions. From this obser-vation, Auracher concluded that hysteresis observed in transition boiling wasdue to the transient nature of the data obtained in previous studies and that nohysteresis existed when the data were obtained under steady-state conditions.The issue is far from being resolved since Auracher did not provide any steady-state transition boiling data with liquids having contact angles vastly differentfrom zero, and a majority of the transition boiling data showing hysteresis wereobtained by other investigators only under relatively slow transient conditions.

Equation 12 has been used to correlate the dependence of wall heat fluxon wall superheat in transition boiling. Generally, empirical expressions fordependence ofql , qv, andαw on1Tare used so that predicted heat fluxes matchthe transition boiling data and the maximum and minimum heat fluxes at theupper and lower end of the transition boiling data. A few semi-mechanisticapproaches (e.g. Shoji 1992) have been proposed for prediction of transitionboiling heat fluxes. Although these models agree with the data reasonably well,independent verification of different submodels that contribute to the overallmodel is lacking.

To facilitate mechanistic modeling of transition boiling, several experimentalstudies have measured the wet area fraction during nucleate boiling (see Dhir1991) and heat transfer associated with liquid contacts (see Chen & Hsu 1995).

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Using a microthermocouple probe, Chen & Hsu (1995) measured the transientsurface heat fluxes when a liquid droplet lands on a hot, initially dry, surface.They noted that the process of heat transfer during a short period of liquidcontact is very complex but found that the average heat flux increases with wallsuperheat. During contact periods the droplets that initially had a subcooling of80◦C yielded average heat fluxes as high as 107 W/m2 at high wall superheats.

More systematic studies similar to that of Chen & Hsu (1995) are neededto understand the physics of heat transfer during transient liquid-solid contactsbefore credible models for transition boiling can be developed. These studiesneed to be supplemented with measurement of position- and time-dependenttemperature and wall void fractions and with visual observations of the structureof the vapor-liquid interface.

FILM BOILING

Film boiling is amenable to straightforward analysis; however, in carrying outthe analysis, many simplifications with respect to the matching of interfacialconditions and the shape of the interface are made. As a result, empiricalconstants must be used to match predictions with the data.

Semi-Mechanistic ModelsSakurai et al (1990a,b) developed comprehensive correlations for saturated andsubcooled film boiling on horizontal cylinders. In developing the correlations,which include the effect of radiation, the functional form of the correlations wasobtained by solving the conservation equations for a two-layered configurationof subcooled film boiling. Sakurai et al also developed a database that covereda large range of system variables such as heater diameter, wall superheat, liquidsubcooling, and pressure. The correlations agreed well with the data, includingthat obtained for cryogenic liquids.

Sakurai & Shiotsu (1992) extended their correlations to include a verticalsurface and a sphere. By measuring the size of the bubble departing a heaterunder saturated film boiling conditions and bubble release frequency, Sakurai &Shiotsu also developed an expression for minimum heat flux on horizontal cylin-ders. In addition, they showed that Lienhard & Wong’s (1964) semi-mechanisticcorrelation tends to overpredict the effect of system pressure, and that the col-lapse of vapor film is influenced by the heater surface temperature rather thanthe heat flux. For saturated film boiling, the minimum film boiling tempera-ture increased with pressure and asymptotically approached the homogeneousnucleation temperature. In correlating the minimum film boiling temperature,Sakurai & Shiotsu (1992) accounted for the reduction in the heater surface tem-perature that occurs upon spontaneous contact of the liquid with the surface.

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Mechanistic StudiesStudies have also focused on understanding the structure of the vapor-liquidinterface in film boiling. Bui & Dhir (1985a) investigated saturated film boilingon a vertical surface. They noted that both long and short waves exist onthe interface. The long waves evolve into large bulges, and vapor from theintervening thin-film region feeds the large bulges. As a result, the flow pathlength is shortened, and higher average heat transfer rates occur than thosepredicted for a continuous flow path. They also noted that a significant variationin local heat transfer coefficient occurs with time as large bulges and thin-filmregions sweep over the heated surface. In a study of subcooled film boilingon a vertical surface, Vijaykumar & Dhir (1992a) noted that, as in saturatedfilm boiling, two types of waves (termed ripples and large waves) exist on theinterface.

Figure 7 shows photographs of the frontal view of film boiling on a verticalsurface. The waves tend to form at a short distance downstream of the leadingedge. The amplitude of the long waves controls the interface velocity. Basedon the amplitude-to-wavelength ratio, the interface behavior in subcooled filmboiling was divided into two regimes. In the low subcooling regime, three-dimensional waves exist on the interface. The flows that result from the de-veloping liquid boundary layer and from the rapid acceleration of the interfacemerge downstream of the leading-edge vapor layer. The fluid in the mergedregion is entrained by the moving interface.

Figure 8 shows profiles of axial velocity in the liquid boundary layer whenwater had a subcooling of 6.7 K. At the top of the first wave peak downstreamof the leading edge, the velocity profile in the liquid shows a steep gradient.In the valley behind the first peak, flow expands and velocities decrease in theboundary layer. A local minimum and a maximum in velocity profile can benoted in the boundary layer. Flow contracted and expanded at the succeedingwave peaks and valleys, respectively. At higher subcoolings, wave structuredegenerates into two-dimensional waves. The liquid boundary layer thins, butthe phenomenon of expansion and contraction of the flow in the valleys and atthe peaks, respectively, persists.

Using a holographic technique, Vijaykumar & Dhir (1992b) also obtainedtemperature profiles and heat fluxes in the liquid layer adjacent to the vaporfilm. Figure 9 shows the interface shape and the heat flux into the liquid atthe interface for water subcooling of 1.6 K. A relatively large heat flux existsat the frontal region of the leading-edge layer (location a). The rate of heattransfer decreases to a minimum value (location f) and thereafter remains fairlyconstant in the valley region. At the frontal region of the first peak, the heatflux increases gradually, reaching a maximum at the top of the peak beforedecreasing again.

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Figure 7 Photographs of frontal view of film boiling on a vertical surface.

From information such as that shown in Figure 9, Vijaykumar & Dhir (1992b)concluded that liquid-side heat transfer over each wavelength shows a cyclicbehavior with the highest heat flux occurring at the peaks and the minimumoccurring in the wave valleys. Invariably, the highest heat flux exceeds the heatflux by conduction through the vapor bulge. High local liquid-side heat fluxat the peaks suggests the possibility of local condensation. Evaporation in thevalleys and condensation at the peaks results in little increase in the substratefilm thickness in the vapor flow direction. The average liquid-side heat transferis enhanced both by the cyclic behavior and by the increased interfacial area.Thus, Vijaykumar & Dhir were able to explain that the underprediction ofsubcooled film boiling heat transfer by the two-layer models that assume aplane interface is largely the result of the underprediction in the liquid-side heatflux.

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Figure 8 Velocity profiles of the component of the velocity (1T = 207 K,1Tsub= 6.7 K) alongthe vertical plate.

Complete Numerical SimulationSon & Dhir (1996) carried out a complete numerical simulation of the evolu-tion of the liquid-vapor interface during saturated film boiling on a horizontalplate. They invoked the assumption of axi-symmetry and considered the cir-cular regions around the nodes and anti-nodes of the Taylor wave. Each of thecircular regions was assumed to have an area equal to half the square of the two-dimensional “most dangerous” Taylor wavelength. Figure 10a shows at 1-atmpressure the calculated shapes of the evolving interface for aβ (≡ cpv1T/h f g)value of 0.09, which corresponds to a wall superheat of 100 K for water.

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Figure 9 Liquid-side heat transfer as a function of distance along the vertical plate for1T = 160K, 1Tsub = 1.6 K.

Figure 10b shows the dependence of Nusselt number on dimensionless timeand dimensionless radial position forβ = 0.09. Most of the heat is transferredin the thin-film region and in the bubble region in the vicinity of the point ofthe minimum film thickness. Little heat transfer takes place under the bubblecore. The magnitude of the highest heat transfer coefficient increases with time,and the location at which the film is the thinnest moves radially inward as theinterface evolves into a bubble.

These observations run counter to Berenson’s (1961) assumption of a uniformheat transfer rate in the thin-film region connecting neighboring bubbles. Themagnitude of the highest heat transfer coefficient increases as the wall superheator β decreases. Since the film is the thinnest where the heat transfer rate is the

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Figure 10 Prediction of film boiling on a horizontal surface. (a) Evolution of the interface.(b) Heat transfer coefficient as a function of position and time. (c) Nusselt number based onarea-averaged heat transfer coefficient.

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Figure 10 (Continued)

highest, the magnitude of the minimum thickness decreases as wall superheatdecreases. At a certain superheat, the film may become so thin that it ruptures.Any perturbations near the interface can accelerate the rupture process. Filmrupture can lead to local liquid-solid contacts and can, in turn, cause the stablefilm boiling to cease. However, the exact nature of these contacts and thespreading behavior can be determined only if a conjugate problem involvingconduction in the solid is solved simultaneously. This was not done in Son &Dhir’s study (1996).

Figure 10c shows the Nusselt numbers based on the heat transfer coefficientaveraged over the cell area. With increase in wall superheat, vapor film thicknessin the thin-film region increases, and as a result, the heat transfer coefficientdecreases. The average Nusselt number also shows some dependence on time,but it is much less than that seen in the local heat transfer coefficient.

The Nusselt numbers based on the area- and time-averaged heat transfer co-efficients obtained by integrating the area under the curves (see Figure 10c) areabout 30 to 35% lower than those obtained from Berenson’s (1961) correlation.Also, the numerically calculated time- and area-averaged heat transfer coeffi-cients are closer to the lower bound of Hosler & Westwater’s (1962) data. Onepossible reason for the underprediction of the heat transfer rate from the numeri-cal simulation is the use of axi-symmetric analysis instead of three-dimensionalanalysis, which is more appropriate for the low-pressure film boiling on a hor-izontal surface.

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Nevertheless, the analysis provides a better physical description of the filmboiling process on a horizontal surface. The numerical analysis such as thatused by Son & Dhir can be a useful experimental tool for investigating theeffect of various system variables on the film boiling process, including thethermal properties of the heater material. Numerical simulation of phase changeprocesses is expected to be used much more in the future.

FLOW BOILING

External Flow BoilingTwo of the most often studied geometries under external flow conditions arethose of a liquid jet impinging on a horizontal surface and of flow normal toa horizontal cylinder. In these two cases, the flow can be along the directionof gravity, against the direction of gravity, or normal to gravity. Fully devel-oped nucleate boiling data obtained under forced flow conditions represent anextension of the pool boiling curve. However, in partial nucleate boiling, thefunctional dependence of the boiling heat flux on wall superheat is weaker underforced flow condition.

Several investigators have developed semi-theoretical correlations for themaximum heat flux obtained with impinging jets. Monde (1987) extended Hara-mura & Katto’s (1983) critical liquid-layer model, whereas Sharan & Lienhard(1985) used the mechanical energy stability criterion initially proposed by Lien-hard & Eichhorn (1979). According to this criterion, the maximum heat fluxoccurs when the rate at which kinetic energy added to the coolant (as a result ofevaporation) exceeds the rate at which energy is consumed in the formation ofnew droplets. During jet impingement cooling, substantially higher maximumheat fluxes are obtained at relatively low jet velocities.

Lay & Dhir (1995b) showed that macro- or micro-modification of the surfacecan enhance twofold to threefold the maximum or critical heat flux during jetimpingement cooling. The macro-modification is in the form of radial channels,which help retain the liquid on the surface. The micro-modifications, in contrast,provide a high density of active nucleation sites at low wall superheats.

Lienhard (1988) reviewed the area of prediction of maximum heat fluxon cylinders. He also gave correlations applicable to gravity-influenced andgravity-free data. Jensen & Hsu (1988) showed that for upflowing cross-flowover tube bundles, nucleate-boiling heat transfer coefficients are influenced bythe location of the tube in the bundle. Because of an accumulation of vaporalong the flow direction, flow regimes change. A liquid film with vapor coreis observed on tubes far away from the inlet. Thus, different types of criticalheat flux mechanisms can prevail in the lower and upper parts of a tube bundle.Jensen (1988) reviewed cross-flow boiling on horizontal tube bundles.

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Internal Flow BoilingExtensive studies of boiling in tubes have been reported in the literature becauseof the need for researchers to understand the cooling limits of nuclear reactorcores and steam generators. As a result of the addition of heat along the axisof the tube, the enthalpy of the liquid entering the tube increases as it flowsthrough the tube. When a subcooled liquid enters the tube, forced convection isfollowed by subcooled boiling at the wall, which in turn gives way to saturatedor bulk boiling. After initiation of bulk boiling, the addition of vapor alongthe tube axis causes flow regimes to change from bubbly flow, to slug flow, toannular flow, and eventually to entrained flow.

In bubbly flows, discrete vapor bubbles exist in the continuous phase (liquid).With an increase in the vapor flow rate, small bubbles merge to form longbubbles separated by liquid filaments. These bubbles occupy almost the entirecross-section of the tube. As the vapor flow rate increases further, the longbubbles merge to give rise to annular flow. In annular flow, liquid is confinedto the thin region adjacent to the wall, whereas vapor occupies the core of thetube. At a still higher vapor flow rate, liquid droplets are entrained in the vaporphase, which is now continuous.

Bergles & Rohsenow (1964), among others, suggested a correlation for par-tial nucleate boiling based on interpolation of the data for forced convection andfully developed nucleate boiling. For fully developed nucleate boiling in forcedflow, the correlations developed for pool boiling are applicable, although cor-relations specific to flow boiling in tubes have also been reported. According tothese correlations, the exponentm in Equation 1 varies between 2 and 4. Thesecorrelations are valid for both bubbly and slug flows. In annular flow, a liquidfilm covers the heated surface. Chen (1963) developed a correlation that ac-counts for nucleate boiling in the liquid film and forced convection cooling of thewall by the film. In entrained flow, correlations for single phase forced flow areapplicable. However, these correlations must be corrected for the presence ofdroplets in the core flow. The droplets generally enhance the wall heat transfer.

For flow in tubes, a distinction must be made between critical heat fluxunder low-flow and high-flow conditions. Under low-flow conditions, the liquidfilm can dry out and lead to a rise in wall temperature as the heat removalrate degrades. Under high-flow conditions, the critical heat flux conditioncorresponds to local dryout of the tube surface even though the tube core isfull of liquid. Upon occurrence of the critical heat flux condition, the tubesurface temperature rises rapidly to a very high value, and the tube can fail ifthe temperature exceeds the melting temperature of the heater material.

Several correlations for dryout and critical heat flux covering different fluids,flow rates, tube diameter, local quality, and system pressure have been reported.

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“Look-up” tables (see e.g. Groenveld & Lueng 1989) have been developed thatrequire less computing time and that can be updated easily as new data becomeavailable.

Mechanistic models for critical heat flux in tubes have also been developed.These models, however, use empirical constants to match the predictions withthe data. Several competing mechanisms for critical heat flux have been pro-posed. Two of the most commonly proposed mechanisms for critical heat fluxare (a) the inability of the core flow to remove vapor generated at the wall,and (b) the formation of a persistent dry patch underneath a bubble attached tothe wall or resulting from dryout of the thin liquid film between the wall anda large bubble. Katto (1994) and Celata et al (1994) provided comprehensivereviews of correlations and models for critical heat flux. Efforts similar to thosedescribed earlier for film boiling are currently being made to predict criticalheat flux by carrying out complete numerical simulation of the process (see e.g.Lahey 1996).

ACKNOWLEDGMENTS

The author appreciates the support from the NASA Microgravity Fluid PhysicsProgram, with Dr. David Chao as Project Scientist. The author is also gratefulto Ms. Cindy Gilbert for skillfully typing the manuscript.

Visit the Annual Reviews home pageathttp://www.AnnualReviews.org.

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